Where are the functions f1(x) = |sin(x)| and f2(x) = sin(|x|) differentiable? (Use n as an arbitrary integer constant.)

Question

Where are the functions f₁(x) = |sin(x)| and f₂(x) = sin(|x|) differentiable? (Use n as an arbitrary integer constant.)

f₁(x) = |sin(x)| is differentiable for all x ≠ 

 

f₂(x) = sin(|x|) is differentiable for all x ≠ 

Answer 1

Hi Kaitlyn,

These functions will not be differentiable everywhere they equal zero. That's because the absolute value function will reflect all of the negative values above the x-axis, resulting in sharp corners (called cusps) at the axis itself. Functions cannot be differentiated at cusps.

For f(x) = sin(x), it will equal zero when x = 0 + π*n, or just x = π*n. |sin(x)| will therefore not be differentiable at x = π*n, where n is some integer constant.

For f(x) = cos(x), it will equal zero when x = π/2 + π*n. |cos(x)| will therefore not be differentiable at x = π/2 + π*n, where n is some integer constant.

Hope this helps!